Optimal. Leaf size=227 \[ 2 a^3 e \sqrt {c+d x}-2 a^3 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]
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Rubi [A] time = 0.26, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {153, 147, 50, 63, 208} \[ \frac {2 (c+d x)^{3/2} \left (2 \left (3 a^2 b d^2 (45 d e-16 c f)+20 a^3 d^3 f-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+2 a^3 e \sqrt {c+d x}-2 a^3 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\frac {2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 147
Rule 153
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 \sqrt {c+d x} (e+f x)}{x} \, dx &=\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 \int \frac {(a+b x)^2 \sqrt {c+d x} \left (\frac {9 a d e}{2}+\frac {3}{2} (3 b d e-2 b c f+2 a d f) x\right )}{x} \, dx}{9 d}\\ &=\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {4 \int \frac {(a+b x) \sqrt {c+d x} \left (\frac {63}{4} a^2 d^2 e+\frac {3}{4} \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{x} \, dx}{63 d^2}\\ &=\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 e\right ) \int \frac {\sqrt {c+d x}}{x} \, dx\\ &=2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 c e\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx\\ &=2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\frac {\left (2 a^3 c e\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=2 a^3 e \sqrt {c+d x}+\frac {2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac {2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}-2 a^3 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 205, normalized size = 0.90 \[ \frac {2 \left (3 d e \left (105 a^3 d^3 \sqrt {c+d x}-105 a^3 \sqrt {c} d^3 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+35 b (c+d x)^{3/2} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )-21 b^2 (c+d x)^{5/2} (2 b c-3 a d)+15 b^3 (c+d x)^{7/2}\right )-f (c+d x)^{3/2} \left (135 b^2 (c+d x)^2 (b c-a d)-189 b (c+d x) (b c-a d)^2+105 (b c-a d)^3-35 b^3 (c+d x)^3\right )\right )}{315 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 649, normalized size = 2.86 \[ \left [\frac {315 \, a^{3} \sqrt {c} d^{4} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}}{315 \, d^{4}}, \frac {2 \, {\left (315 \, a^{3} \sqrt {-c} d^{4} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (35 \, b^{3} d^{4} f x^{4} + 5 \, {\left (9 \, b^{3} d^{4} e + {\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e - {\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \, {\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e - {\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f - {\left (3 \, {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e - {\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.41, size = 338, normalized size = 1.49 \[ \frac {2 \, a^{3} c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{\sqrt {-c}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} d^{32} f - 135 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} c d^{32} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c^{2} d^{32} f - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{3} d^{32} f + 135 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{2} d^{33} f - 378 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} c d^{33} f + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{33} f + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b d^{34} f - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b c d^{34} f + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} d^{35} f + 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{33} e - 126 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{33} e + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{33} e + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{34} e - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{34} e + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{35} e + 315 \, \sqrt {d x + c} a^{3} d^{36} e\right )}}{315 \, d^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 301, normalized size = 1.33 \[ \frac {-2 a^{3} \sqrt {c}\, d^{4} e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+2 \sqrt {d x +c}\, a^{3} d^{4} e +\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3} d^{3} f}{3}-2 \left (d x +c \right )^{\frac {3}{2}} a^{2} b c \,d^{2} f +2 \left (d x +c \right )^{\frac {3}{2}} a^{2} b \,d^{3} e +2 \left (d x +c \right )^{\frac {3}{2}} a \,b^{2} c^{2} d f -2 \left (d x +c \right )^{\frac {3}{2}} a \,b^{2} c \,d^{2} e -\frac {2 \left (d x +c \right )^{\frac {3}{2}} b^{3} c^{3} f}{3}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} b^{3} c^{2} d e}{3}+\frac {6 \left (d x +c \right )^{\frac {5}{2}} a^{2} b \,d^{2} f}{5}-\frac {12 \left (d x +c \right )^{\frac {5}{2}} a \,b^{2} c d f}{5}+\frac {6 \left (d x +c \right )^{\frac {5}{2}} a \,b^{2} d^{2} e}{5}+\frac {6 \left (d x +c \right )^{\frac {5}{2}} b^{3} c^{2} f}{5}-\frac {4 \left (d x +c \right )^{\frac {5}{2}} b^{3} c d e}{5}+\frac {6 \left (d x +c \right )^{\frac {7}{2}} a \,b^{2} d f}{7}-\frac {6 \left (d x +c \right )^{\frac {7}{2}} b^{3} c f}{7}+\frac {2 \left (d x +c \right )^{\frac {7}{2}} b^{3} d e}{7}+\frac {2 \left (d x +c \right )^{\frac {9}{2}} b^{3} f}{9}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 239, normalized size = 1.05 \[ a^{3} \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{3} d^{4} e + 35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{3} f + 45 \, {\left (b^{3} d e - 3 \, {\left (b^{3} c - a b^{2} d\right )} f\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 63 \, {\left ({\left (2 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} e - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} f\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{315 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 413, normalized size = 1.82 \[ \left (c\,\left (c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{d^4}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (c\,f-d\,e\right )}{d^4}\right )\,\sqrt {c+d\,x}+\left (\frac {c\,\left (c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{d^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d\,f-4\,b\,c\,f+3\,b\,d\,e\right )}{3\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}+\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{7\,d^4}+\frac {2\,b^3\,c\,f}{7\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\left (\frac {c\,\left (\frac {2\,b^3\,d\,e-8\,b^3\,c\,f+6\,a\,b^2\,d\,f}{d^4}+\frac {2\,b^3\,c\,f}{d^4}\right )}{5}+\frac {6\,b\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-2\,b\,c\,f+b\,d\,e\right )}{5\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,b^3\,f\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+a^3\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.79, size = 274, normalized size = 1.21 \[ \frac {2 a^{3} c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a^{3} e \sqrt {c + d x} + \frac {2 b^{3} f \left (c + d x\right )^{\frac {9}{2}}}{9 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {7}{2}} \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}} \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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